Why do people like polytopes?

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atyy
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The first and third paper are related to string theory.

http://arxiv.org/abs/0905.3627
Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine

http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Dona', Simone Speziale

http://arxiv.org/abs/1012.6030
A Note on Polytopes for Scattering Amplitudes
Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Andrew Hodges, Jaroslav Trnka
 

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  • #2
marcus
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atyy
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The relation of the polytopes and space is much more subtle in string theory. I wonder if LQG can learn something from that.
 
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atyy
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Here is the Barbieri's 1997 paper about tetrahedra in LQG, which is cited by Krasnov's paper studying twistors and string theory. Krasnov's work was in turn studied by Witten, who in footnote 13 comments on the indirect relation between his work and Krasnov's.

http://arxiv.org/abs/gr-qc/9707010
Quantum tetrahedra and simplicial spin network
A.Barbieri

http://arxiv.org/abs/hep-th/0311162
Twistors, CFT and Holography
Kirill Krasnov (AEI, Golm/Potsdam)

http://arxiv.org/abs/hep-th/0312171
Perturbative Gauge Theory As A String Theory In Twistor Space
Edward Witten

However, that paper by Witten seems only indirectly related to the current state of that twisor minirevolution. And Motl also points out that off-shell gauge theory isn't yet in play: "And it could perhaps shed some new light of the "full string theory" including AdS5 x S5 gravity (e.g. off-shell gauge theory - note that only on-shell gauge theory is studied in this whole business so far) and perhaps even more general mysteries." http://motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html
 
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marcus
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Atyy,
Suppose we look at two of the three papers in your OP and consider the original question as simply as possible "why do people like polytopes?" We can put the question more specifically, in the case of these two papers, as "Why do these 6 people like polyhedra?"It may give us some clues as to where Freidel Speziale Livine etc will be going with the new "Principle of Relative Locality". Most of these 6 people are repeat co-authors who must certainly share ideas. Both about polyhedra and about "relative locality" (a new initiative as yet only partially grasped.)

...

http://arxiv.org/abs/0905.3627
Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine

http://arxiv.org/abs/1009.3402
Polyhedra in loop quantum gravity
Eugenio Bianchi, Pietro Dona', Simone Speziale
...
...
I guess the idea of a "space of shapes" is basic. Spinnetwork nodes are intertwiners--the space of intertwiners is a hilbert space of (quantum) shapes. Roughly speaking. A quantum geometry should be built by gluing together quantum (uncertain, indefinite) shapes. This is not rigorous, just trying to sense why the idea of quantumpolyhedra is so basic. Let's see what Freidel Krasnov Livine (FKL) say:

========quote FKL=========
Our discussion has so far been quite mathematical, so we would now like to switch to a more heuristic description and explain the significance of our results for the field of quantum gravity. As we have already mentioned, the n = 4 intertwiner that we have characterized in this paper in most details plays a very important role in both the loop quantum gravity and the spin foam approaches. These intertwiners have so far been characterized using the real basis... In particular, the main building blocks of the spin foam models – the (15j)-symbols and their analogs – arise as simple pairings of 5 of such intertwiners (for some choice of the channels ij). The main result of this paper is a holomorphic description of the space of intertwiners, and, in particular, an explicit basis in Hj1,...,j4 given by the holomorphic intertwiner... While the basis ..., being discrete, may be convenient for some purposes, the underlying geometric interpretation in
it is quite hidden. Indeed, recalling the interpretation of the intertwiners from Hj1,...,j4 as giving the states of a quantum tetrahedron, the states...describe a tetrahedron whose shape is maximally uncertain. In contrast, the intertwine..., being holomorphic, are coherent states in that they manage to contain the complete information about the shape of the tetrahedron coded into the real and imaginary parts of the cross-ratio coordinate Z. We give an explicit description of this in the main text.
Thus, with the holomorphic intertwiner.... at our disposal, we can now characterize the “quantum geometry” much more completely than it was possible before. Indeed, we can now build the spin networks – states of quantum geometry – using the holomorphic intertwiners. The nodes of these spin networks then receive a well-defined geometric interpretation as corresponding to particular tetrahedral shapes. Similarly, the spin foam model simplex amplitudes can now be built using the coherent intertwiners, and then the basic object becomes not the (15j)-symbol of previous studies, but the (10j)-(5Z)-symbol with a well-defined geometrical interpretation. Where this will lead the subjects of loop quantum gravity and spin foams remains to be seen, but the very availability of this new technology opens way to many new developments and, we hope, will give a new impetus to the field that is already very active after the introduction of the new spin foam models in [11–15].
The organization of this paper is as follows. In section II we describe how the phase space that we would like to quantize arises as a result of the symplectic reduction of a simpler phase space.... semi-classical limit of large spins and show that it takes the form precisely as is expected from the point of view of geometric quantization. ...
===endquote===
 
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  • #6
atyy
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It may give us some clues as to where Freidel Speziale Livine etc will be going with the new "Principle of Relative Locality".
Yes.

I have a hunch the relative locality paper is way too conservative.

But FKL have a different idea of where LQG should go - they don't know quite where yet - so if you read their papers, there's still plenty of standard LQG interpretation.

But I have the same feeling as tom.stoer on another thread where he said you cannot just add matter on top. Actually, that probably means that even the interpretation of spacetime is not correct. Of course I have no idea in technical terms what the dramatically new interpretation should be. But that is the lesson of AdS/CFT.

If you notice FL are on that paper where they get some sort of unification, getting noncommutative field theory form 3D gravity.

And K now has two papers that mention LQG and AdS/CFT trying to link them.

Cachazo is at Perimeter, so there's plenty of twistor stuff in the air there.

Although I'm not sure relative locality is the solution, F is working on it because he would like noncommutative field theory to play a role somewhere in this unification view, and noncommutative field theory is related somehow to DSR.

K, as you know, has great instinct.

While I'm trying to read (Smolin's?) subconscious ideas, let me point out that Vidal is joining the Perimeter faculty. His work with Singh hints at a link between spin networks and AdS/CFT.
 
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marcus
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to augment a little what you said. the basic idea of rel. loc. is that physics happens in phase space, not spacetime. there is no abs. spacetime. different observers project down from the 8d phasespace to the 4d spacetime differently. they have different slices.

primitively, think of phasespace as 4d of position and 4d of momentum and the momentum space is *curved* so that there is a funny algebra there, when you try to add vectors they do not add simply like in a flat vectorspace.

so these people may be driven by their ideas to construct an 8d geometric object perhaps consisting of a 4d polyhedron + another 4d polyhedron.

also they could be wrong. the "relative locality" paper said it could make predictions about physics and be testable and falsifiable.

If tested and found wrong that would be so nice because then we would still have an absolute 4d spacetime, and momentumspace would still be flat. This is much much simpler, so we could be very happy if the Freidel thing is falsified.

but in any case they have the project underway, so this is how I will see what they are doing with polyhedra. It is very relevant to plain 4D spinfoam QG, but it is also potentially relevant (I suspect) to the quantum geometry of a 4D+4D phase space.
Just a guess, could easily be wrong.

I think Freidel talks about "rel. loc." in the conference-call seminar on March 1, 6 weeks hence. Rovelli may ask him some questions.
 
  • #8
atyy
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No, I don't want my 4D spacetime. Rel loc is a step away - so that's good - I suspect one must step even further away. Honestly, marcus, when you have a nice meditative evening by the lake, does space seem 3D to you? (Well, obviously that's not a technical question - but 3D space has never been intuitive to me.)
 
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marcus
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:biggrin:

:wink:
 

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